\(\bullet \)  On the weighted \(L^p\) -boundedness. We shall consider \(T_{g^+_1}\)  as a model case. The theory of Calderón–Zygmund operators shows \(T_{\widetilde{k}_1},T_{\widetilde{k}_2^\pm }\in \mathbb {B}(L^p(w_p))\cap \mathbb {B}(L^1(w_1),L^{1,\infty }(w_1))\)  for any \(w_p\in A_p\) . Moreover, recent deep results by [1]} for \(1<p<\infty \)  and by [2]} for \(p=1\)  imply
\({\Vert T_{\widetilde{k}_1}\Vert }_{L^p(w_p)\rightarrow L^p(w_p)}+{\Vert T_{\widetilde{k}_2^\pm }\Vert }_{L^p(w_p)\rightarrow L^p(w_p)}&\lesssim [w_p]_{A_p}^{\max \lbrace 1,1/(p-1)\rbrace },\quad 1<p<\infty ,\\{\Vert T_{\widetilde{k}_1}\Vert }_{L^1(w_1)\rightarrow L^{1,\infty }(w_1)}+{\Vert T_{\widetilde{k}_2^\pm }\Vert }_{L^1(w_1)\rightarrow L^{1,\infty }(w_1)}&\lesssim [w_1]_{A_1}(1+\log [w]_{A_1}).\) 
